3.2176 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=122 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{e \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac{e \log (d+e x)}{a e^2-b d e+c d^2} \]

[Out]

-(((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d
^2 - b*d*e + a*e^2))) + (e*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2) - (e*Log[a + b*
x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.215604, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{e \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac{e \log (d+e x)}{a e^2-b d e+c d^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*x + c*x^2)),x]

[Out]

-(((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d
^2 - b*d*e + a*e^2))) + (e*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2) - (e*Log[a + b*
x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.5278, size = 112, normalized size = 0.92 \[ \frac{e \log{\left (d + e x \right )}}{a e^{2} - b d e + c d^{2}} - \frac{e \log{\left (a + b x + c x^{2} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x+a),x)

[Out]

e*log(d + e*x)/(a*e**2 - b*d*e + c*d**2) - e*log(a + b*x + c*x**2)/(2*(a*e**2 -
b*d*e + c*d**2)) + (b*e - 2*c*d)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(sqrt(-4
*a*c + b**2)*(a*e**2 - b*d*e + c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.144003, size = 105, normalized size = 0.86 \[ \frac{e \sqrt{4 a c-b^2} (\log (a+x (b+c x))-2 \log (d+e x))+(2 b e-4 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)),x]

[Out]

((-4*c*d + 2*b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*e*
(-2*Log[d + e*x] + Log[a + x*(b + c*x)]))/(2*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b
*d - a*e)))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 168, normalized size = 1.4 \[{\frac{e\ln \left ( ex+d \right ) }{a{e}^{2}-bde+c{d}^{2}}}-{\frac{e\ln \left ( c{x}^{2}+bx+a \right ) }{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}}-{\frac{be}{a{e}^{2}-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{cd}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x+a),x)

[Out]

e*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)-1/2*e*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)-1/(a*e
^2-b*d*e+c*d^2)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e+2/(a*e
^2-b*d*e+c*d^2)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*d

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.482295, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, c d - b e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (e \log \left (c x^{2} + b x + a\right ) - 2 \, e \log \left (e x + d\right )\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (2 \, c d - b e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (e \log \left (c x^{2} + b x + a\right ) - 2 \, e \log \left (e x + d\right )\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="fricas")

[Out]

[-1/2*((2*c*d - b*e)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x + (2*c^2*x^2 + 2
*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) + sqrt(b^2 - 4*a*c)*
(e*log(c*x^2 + b*x + a) - 2*e*log(e*x + d)))/((c*d^2 - b*d*e + a*e^2)*sqrt(b^2 -
4*a*c)), 1/2*(2*(2*c*d - b*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a
*c)) - sqrt(-b^2 + 4*a*c)*(e*log(c*x^2 + b*x + a) - 2*e*log(e*x + d)))/((c*d^2 -
b*d*e + a*e^2)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x+a),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207667, size = 170, normalized size = 1.39 \[ -\frac{e{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} + \frac{e^{2}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="giac")

[Out]

-1/2*e*ln(c*x^2 + b*x + a)/(c*d^2 - b*d*e + a*e^2) + e^2*ln(abs(x*e + d))/(c*d^2
*e - b*d*e^2 + a*e^3) + (2*c*d - b*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c
*d^2 - b*d*e + a*e^2)*sqrt(-b^2 + 4*a*c))